<< problem 38 - Pandigital multiples | Champernowne's constant - problem 40 >> |
Problem 39: Integer right triangles
(see projecteuler.net/problem=39)
If p is the perimeter of a right angle triangle with integral length sides, \{ a,b,c \}, there are exactly three solutions for p = 120.
\{ 20,48,52 \}, \{ 24,45,51 \}, \{ 30,40,50 \}
For which value of p <= 1000, is the number of solutions maximised?
My Algorithm
Euclid's formula generates all triplets \{ a,b,c \}, see en.wikipedia.org/wiki/Pythagorean_triple
Assuming a <= b <= c:
a = k * (m^2 - n^2)
b = k * 2mn
c = k * (m^2 + n^2)
perimeter = a + b + c
Integer numbers m, n, k produce all triplets under these conditions:
- m and n are coprime → their Greatest Common Divisor is 1
- m and n are not both odd
a must be positive (as well as b and c) therefore m > n
Furthermore:
perimeter = k * (m^2 - n^2) + k * 2mn + k * (m^2 + n^2)
= k * (m^2 - n^2 + 2mn + m^2 + n^2)
= k * (2m^2 + 2mn)
= 2km * (m+n)
which gives an approximation of the upper limit: 2m^2 < MaxPerimeter
My program evaluates all combinations of m and n. For each valid pair all k are enumerated,
such that the perimeter does not exceed the maximum value.
A simple lookup container
count
stores for each perimeter the number of triangles.Following this precomputation step I perform a second step:
extract those perimeters with more triangles than any smaller perimeter.
The value stored at
best[perimeter]
equals the highest count[i]
for all i <= perimeter
.The actual test cases are plain look-ups into
best
.
Interactive test
You can submit your own input to my program and it will be instantly processed at my server:
This is equivalent toecho "1 120" | ./39
Output:
Note: the original problem's input 1000
cannot be entered
because just copying results is a soft skill reserved for idiots.
(this interactive test is still under development, computations will be aborted after one second)
My code
… was written in C++11 and can be compiled with G++, Clang++, Visual C++. You can download it, too.
#include <iostream>
#include <set>
#include <vector>
// greatest common divisor
unsigned int gcd(unsigned int a, unsigned int b)
{
while (a != 0)
{
unsigned int c = a;
a = b % a;
b = c;
}
return b;
}
int main()
{
const unsigned int MaxPerimeter = 5000000;
// precomputation step 1:
// count all triplets per perimeter (up to upper limit 5 * 10^6)
// [perimeter] => [number of triplets]
std::vector<unsigned int> count(MaxPerimeter + 1, 0);
// note: long long instead of int because otherwise the squares m^2, n^2, ... might overflow
for (unsigned long long m = 1; 2*m*m < MaxPerimeter; m++)
for (unsigned long long n = 1; n < m; n++)
{
// make sure all triplets a,b,c are unique
if (m % 2 == 1 && n % 2 == 1)
continue;
if (gcd(m, n) > 1)
continue;
unsigned int k = 1;
while (true)
{
// see Euclidian formula above
auto a = k * (m*m - n*n);
auto b = k * 2*m*n;
auto c = k * (m*m + n*n);
k++;
// abort if largest perimeter is exceeded
auto perimeter = a + b + c;
if (perimeter > MaxPerimeter)
break;
// ok, found a triplet
count[perimeter]++;
}
}
// precomputation step 2:
// store only best perimeters
unsigned long long bestCount = 0;
std::set<unsigned int> best;
best.insert(0); // degenerated case
for (unsigned int i = 0; i < count.size(); i++)
if (bestCount < count[i])
{
bestCount = count[i];
best.insert(i);
}
// processing input boils down to a simple lookup
unsigned int tests;
std::cin >> tests;
while (tests--)
{
unsigned int maxPerimeter;
std::cin >> maxPerimeter;
// find the perimeter with the largest count
auto i = best.upper_bound(maxPerimeter);
// we went one step too far
i--;
// print result
std::cout << *i << std::endl;
}
return 0;
}
This solution contains 10 empty lines, 15 comments and 3 preprocessor commands.
Benchmark
The correct solution to the original Project Euler problem was found in 0.11 seconds on an Intel® Core™ i7-2600K CPU @ 3.40GHz.
Peak memory usage was about 21 MByte.
(compiled for x86_64 / Linux, GCC flags: -O3 -march=native -fno-exceptions -fno-rtti -std=gnu++11 -DORIGINAL
)
See here for a comparison of all solutions.
Note: interactive tests run on a weaker (=slower) computer. Some interactive tests are compiled without -DORIGINAL
.
Changelog
February 25, 2017 submitted solution
April 18, 2017 added comments
Hackerrank
see https://www.hackerrank.com/contests/projecteuler/challenges/euler039
My code solves 7 out of 7 test cases (score: 100%)
Difficulty
Project Euler ranks this problem at 5% (out of 100%).
Hackerrank describes this problem as easy.
Note:
Hackerrank has strict execution time limits (typically 2 seconds for C++ code) and often a much wider input range than the original problem.
In my opinion, Hackerrank's modified problems are usually a lot harder to solve. As a rule thumb: brute-force is rarely an option.
Links
projecteuler.net/thread=39 - the best forum on the subject (note: you have to submit the correct solution first)
Code in various languages:
C# www.mathblog.dk/project-euler-39-perimeter-right-angle-triangle/ (written by Kristian Edlund)
C github.com/eagletmt/project-euler-c/blob/master/30-39/problem39.c (written by eagletmt)
Java github.com/nayuki/Project-Euler-solutions/blob/master/java/p039.java (written by Nayuki)
Javascript github.com/dsernst/ProjectEuler/blob/master/39 Integer right triangles.js (written by David Ernst)
Go github.com/frrad/project-euler/blob/master/golang/Problem039.go (written by Frederick Robinson)
Mathematica github.com/nayuki/Project-Euler-solutions/blob/master/mathematica/p039.mathematica (written by Nayuki)
Haskell github.com/nayuki/Project-Euler-solutions/blob/master/haskell/p039.hs (written by Nayuki)
Scala github.com/samskivert/euler-scala/blob/master/Euler039.scala (written by Michael Bayne)
Perl github.com/gustafe/projecteuler/blob/master/039-Integer-right-triangles.pl (written by Gustaf Erikson)
Those links are just an unordered selection of source code I found with a semi-automatic search script on Google/Bing/GitHub/whatever.
You will probably stumble upon better solutions when searching on your own.
Maybe not all linked resources produce the correct result and/or exceed time/memory limits.
Heatmap
Please click on a problem's number to open my solution to that problem:
green | solutions solve the original Project Euler problem and have a perfect score of 100% at Hackerrank, too | |
yellow | solutions score less than 100% at Hackerrank (but still solve the original problem easily) | |
gray | problems are already solved but I haven't published my solution yet | |
blue | solutions are relevant for Project Euler only: there wasn't a Hackerrank version of it (at the time I solved it) or it differed too much | |
orange | problems are solved but exceed the time limit of one minute or the memory limit of 256 MByte | |
red | problems are not solved yet but I wrote a simulation to approximate the result or verified at least the given example - usually I sketched a few ideas, too | |
black | problems are solved but access to the solution is blocked for a few days until the next problem is published | |
[new] | the flashing problem is the one I solved most recently |
I stopped working on Project Euler problems around the time they released 617.
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I scored 13526 points (out of 15700 possible points, top rank was 17 out of ≈60000 in August 2017) at Hackerrank's Project Euler+.
My username at Project Euler is stephanbrumme while it's stbrumme at Hackerrank.
Look at my progress and performance pages to get more details.
Copyright
I hope you enjoy my code and learn something - or give me feedback how I can improve my solutions.
All of my solutions can be used for any purpose and I am in no way liable for any damages caused.
You can even remove my name and claim it's yours. But then you shall burn in hell.
The problems and most of the problems' images were created by Project Euler.
Thanks for all their endless effort !!!
<< problem 38 - Pandigital multiples | Champernowne's constant - problem 40 >> |